Factorization conjecture for two-component skew diagrams

Let $bbig mu$ be a rectangular partition, and let $bbig lambda eq\remote$ be a partition containing $bbig mu$ such that the skew diagram $bbig lambda/bbig mu$ has two connected components, represented by partitions $bbig phi$ and $bbig psi$. Let $bbig nu$ satisfy

$$c^\lambda_{\mu,\nu}=1.$$

Write $g^bbig \lambda_{bbig mu,bbig nu}$ and $g^bbig nu_{bbig \phi,bbig \psi}$ for the associated Jack $g$-polynomials, and let $c_{bbig pi,s}$ and $c'_{bbig pi,s}$ be the two hook factors.

Two-component factorization conjecture. The ratio

$$\frac{g^\lambda_{\mu,\nu}}{g^\nu_{\phi,\psi}}=\prod_{s\in\mu}c^*_{\lambda,s}c^*_{\mu,s}$$

d decomposes into linear factors compatibly with the factorizability conjecture, where for $bbig pi=bbig \lambda,bbig mu$ and $s\inbbig mu$, either $c^*_{bbig pi,s}=c_{bbig pi,s}$ or $c^*_{bbig pi,s}=c'_{bbig pi,s}$, and each choice occurs $|bbig mu|$ times in total.

This is a proposed special-case factorization extending the general conjecture to a ratio associated with two connected components. It is not established in the source and remains open.